The size of an intertwine
An intertwine of two graphs H and H′ is a graph G such that G contains both H and H′ as minors, but no proper minor of G contains both H and H′ as minors. We give an upper bound on the size of an intertwine of two given planar graphs. For two planar graphs H and H′, the bound is triply exponential in O(m5) where m≤max(¦V(H)¦, ¦V(H′)¦). We also give an upper on the size of an intertwine of two given trees T and T′. This bound is exponential in O(m3 log m) where m≤max(¦V(T)¦, ¦V(T′)¦). Let O1 be the set of obstructions for a minor closed family L1 and O2 the set of obstructions for a minor closed family L2. It is a natural to ask the following question: how do we given O1 and O2 compute the obstructions for L1∩L2 and L1 ∪ L2. Both these sets of obstructions are known to be finite, and to obtain the obstructions for L1∩ L2 is easy. However, to compute the obstructions for L1∪ L2 is hard. Our upper bound enables us to given O1 and O2 compute a bound on the size of any obstruction for L1 ∪ L2 whenever L1 and L2 are families of planar graphs.
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